OF '0

TO"!'!.L OF '0 F~~CFS ON!.:!

TV! A Y B~ XEROXED

~Wil~oUI AUlhor's "e .. miss~on)

ASYMPTOTIC BEHAVIOR AND TRAVELING WAVES FOR SOME

POPULATION MODELS

by

@Dashun Xu

A thesis sublllitted to the School of Graduate Studies

in partial fuifillllent of the requirelllents for the degree of

Doctor of Philosophy

Departlllent of Mathelllatics and Statistics Melllorial University of Newfoundland

March 2004 Sublllitted

St. John's Newfoundland

DEC 052005

Canada

Abstract

Since the 1970s, more and more mathematicians have been trying to propose reason- able models for the growth of species in all kinds of environments and for the spread of epidemic diseases, and to understand the long-term behavior of their modelling systems. This thesis, consisting of five chapters, mainly deals with the dynamics of population and epidemic models represented by some time-delayed ordinary and partial differential equations, and reaction-diffusion systems.

In Chapter 1, we present some basic concepts and theorems, which involve the

theories of monotone dynamics, uniform persistence, essential spectrum of linear operators, asymptotic speeds of spread and minimal traveling wave speed.

Based on some specific competitive models, we formulate in Chapter 2 a class of asymptotically periodic delay differential equations, which models multi-species competition, and investigate the global dynamics of the model. More precisely, we established the sufficient conditions for competitive coexistence, exclusion and uniform persistence via theories of competitive systems on Banach spaces, uniform persistence, periodic and asymptotically periodic semiflows.

Chapter 3 focuses on a nonlocal reaction-diffusion equation modelling the growth

11

Since the 1970s, more and more mathematicians have been trying to propose reason- able models for the growth of species in all kinds of environments and for the spread of epidemic diseases, and to understand the long-term behavior of their modelling systems. This thesis, consisting of five chapters, mainly deals with the dynamics of population and epidemic models represented by some time-delayed ordinary and partial differential equations, and reaction-diffusion systems.

In Chapter 1, we present some basic concepts and theorems, which involve the theories of monotone dynamics, uniform persistence, essential spectrum of linear operators, asymptotic speeds of spread and minimal traveling wave speed.

Based on some specific competitive models, we formulate in Chapter 2 a class of asymptotically periodic delay differential equations, which models multi-species competition, and investigate the global dynamics of the model. More precisely, we established the sufficient conditions for competitive coexistence, exclusion and uniform persistence via theories of competitive systems on Banach spaces, uniform persistence, periodic and asymptotically periodic semifiows.

Chapter 3 focuses on a nonlocal reaction-diffusion equation modelling the growth

11

Acknow ledgernents

I would like to express my great appreciation to my supervisor, Professor Xiaoqiang Zhao, for his careful guidance, unbelievable patience and invaluable suggestions throughout my Ph.D. program. Without his help, this work would not have been possible. My deepest respect to him is not only for his kind encouragement and brilliant insight, but also for his great effort of training me to be a mathematical researcher and serving me as a model of very hard-working and productive scholar.

I wish to express sincere thanks to Professors Peter Booth, Hermann Brunner, Andy Foster and Xingfu Zou for teaching me the theories of topology, numerical solutions of partial differential equations, dynamical systems and delay differential equations. I also thank them for their helpful discussions, frequent advice and encouragement. In addition, I would like to thank Dr. Bruce Watson and Mrs.

Gaskill for helping me in comprehensive tests and oral English, respectively.

I would like to take this opportunity to thank the NSERC of Canada, the School of Graduate Studies and Department of Mathematics and Statistics, headed by Dr.

Herbert S. Gaskill and currently by Dr. Bruce Watson, for providing me financial support and very convenient facilities. Thanks also goes to all staff members at the

IV

attractivity of a positive steady state. We also discuss the effects of spatial dispersal and maturation period on the evolutionary behavior in two specific cases. Our numerical investigation seems to suggest that the model admits a unique positive steady state even without monotonicity conditions.

In Chapter 4, we consider an epidemic model represented by a reaction-diffusion equation coupled with an ordinary differential equation, which is proposed by Ca- passo et al. Here, the existence, uniqueness (up to translation) and global exponen- tial stability with phase shift of bistable traveling waves are studied by phase plane techniques, monotone semiflow approaches and a detailed spectrum analysis.

In Chapter 5, the asymptotic speeds of spread for solutions and traveling wave solutions to the integral version of the epidemic model in Chapter 4 are investigated.

Our results show that the minimal wave speed for monotone traveling waves coin- cides with the asymptotic speed of spread for solutions with initial functions having compact supports. Some numerical simulations are also provided.

111

Acknow-ledgernents

I would like to express my great appreciation to my supervisor, Professor Xiaoqiang Zhao, for his careful guidance, unbelievable patience and invaluable suggestions throughout my Ph.D. program. Without his help, this work would not have been possible. My deepest respect to him is not only for his kind encouragement and brilliant insight, but also for his great effort of training me to be a mathematical researcher and serving me as a model of very hard-working and productive scholar.

I wish to express sincere thanks to Professors Peter Booth, Hermann Brunner, Andy Foster and Xingfu Zou for teaching me the theories of topology, numerical solutions of partial differential equations, dynamical systems and delay differential equations. I also thank them for their helpful discussions, frequent advice and encouragement. In addition, I would like to thank Dr. Bruce Watson and Mrs.

Gaskill for helping me in comprehensive tests and oral English, respectively.

I would like to take this opportunity to thank the NSERC of Canada, the School of Graduate Studies and Department of Mathematics and Statistics, headed by Dr.

Herbert S. Gaskill and currently by Dr. Bruce Watson, for providing me financial support and very convenient facilities. Thanks also goes to all staff members at the

IV

I reserve special recognition for my companions at MUN. Yi Liu has experienced almost every high and low point of my studies and personal life right along with me. Jingtang Ma, Fang Zhang and Jiajia Zhang demonstrated understanding .and a willingness to help whenever possible. Dr. Shengqiang Liu has shared with me his excellent insights into population biology. Thanks to my other friends for their support, help and understanding. Here, I would like to mention some of them: Dr.

Hongjun Cao, Dr. Yu Chang, Dr. Zhiyuan Jia, Mr. Lei Li, Dr. Jingliang Wang, Dr. Lin Wang, Dr. Ruiqi Wang, Mr. Haiyan Yang, Dr. Yuan Yuan and Mr. Yubo Zou. I hope that I have made half the positive impact on all of them that they have made on me.

lt is my great pleasure to thank Professors Zhujun Jing, Dazhi Meng and Jian- hong Wu for their constant encouragement and frequent discussions. Without them, I could never have studied abroad.

I am deeply indebted to my parents. Their understanding and emotional support have been inspiring me to complete my studies. Also, a big thank you goes to my sisters and brothers. From the bottom of my heart, I am proud of all of them.

v

department for their kind help.

I reserve special recognition for my companions at MUN. Yi Liu has experienced almost every high and low point of my studies and personal life right along with me. Jingtang Ma, Fang Zhang and Jiajia Zhang demonstrated understanding .and a willingness to help whenever possible. Dr. Shengqiang Liu has shared with me his excellent insights into population biology. Thanks to my other friends for their support, help and understanding. Here, I would like to mention some of them: Dr.

Hongjun Cao, Dr. Yu Chang, Dr. Zhiyuan Jia, Mr. Lei Li, Dr. Jingliang Wang, Dr. Lin Wang, Dr. Ruiqi Wang, Mr. Haiyan Yang, Dr. Yuan Yuan and Mr. Yubo Zou. I hope that I have made half the positive impact on all of them that they have made on me.

lt is my great pleasure to thank Professors Zhujun Jing, Dazhi Meng and Jian- hong Wu for their constant encouragement and frequent discussions. Without them, I could never have studied abroad.

I am deeply indebted to my parents. Their understanding and emotional support have been inspiring me to complete my studies. Also, a big thank you goes to my sisters and brothers. From the bottom of my heart, I am proud of all of them.

v

Abstract

Acknowledgements

1 Preliminaries

1.1 Uniform Persistence . . . 1.2 Monotone Dynamical Systems 1.3 Essential Spectrum . . .

1.4 Spreading Speeds and Traveling Waves

2 An Asymptotically Periodic Competitive Model 2.1 Introduction . . . .

2.2 Scalar Delay Differential Equations 2.3 Two-species Competition

2.3.1 The Periodic Case

2.3.2 The Asymptotically Periodic Case.

2.4 Multi-species Competition ..

VI

ii

iv

1 1 3 7 9 15

16

19

28 28 46 50

3 A Nonlocal and Delayed Reaction-Diffusion Model 58

3.1 The Model .. . . . . ... 59

3.2 Existence of Global Attractor 62

3.3 Threshold Dynamics and Positive Steady State 66 3.4 Discussion . . .... . .. . . . . ... 72

4 Bistable Traveling Waves in an Epidemic Model 84 4.1 Introduction . . . . . 85

4.2 Existence of Traveling Waves 88

4.3 Attractivity and Uniqueness 99

4.4 Global Exponential Stability. · 110

4.5 Numerical Simulations .... _{· 118}

5 Spreading Speed and Traveling Waves for a Nonlocal Epidemic Model

5.1 Introduction.

5.2 The Asymptotic Speed of Spread 5.3 Traveling Wave Solutions. . . . .

Bibliography

Vll

121

· 122

· 124

· 132

141

Abstract

Acknowledgelllents

1 Prelilllinaries

1.1 Uniform Persistence .

1.2 Monotone Dynamical Systems 1.3 Essential Spectrum

1.4 Spreading Speeds and Traveling Waves

2 An ASYlllPtotically Periodic COlllpetitive Model

2.1 Introduction . . . . 2.2 Scalar Delay Differential Equations 2.3 Two-species Competition

2.3.1 The Periodic Case

2.3.2 The Asymptotically Periodic Case.

2.4 Multi-species Competition . . . .

VI

ii

iv

1 1 3 7 9

15

16 19

28 28 46 50

List of Figures

3.1 The solution of Example 2 in the case of the Neumann boundary condition. The parameters of the system are as follows:

n =

(0,7r), p

=

q

=

^/-Lj

=

1, dm

=

0.5,

f3 =

0.2, dj

=

0.25, T

=

1.7, ¢(t, x) ⁼

1 - cos(2x). . . .. 78 3.2 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for ^T

=

1, ¢(t, x)

=

1 - cos(2x). . . .. 78 3.3 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for T

=

0.3, ¢(t, x)

=

1 - cos(2x). . . .. 79 3.4 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for ^T

=

0.3, ¢(t, x)

=

3 - 3 cos(2x). . . .. 79 3.5 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for T

=

^{0.3, ¢(t, x)}

=

^{1 -} cos(4x). . . .. 80 3.6 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for ^T

=

0.3, ¢(t, x)

=

5 - 5cos(4x). . . . . . 80

Vlll

3.1 The Model. . . . 3.2 Existence of Global Attractor

3.3 Threshold Dynamics and Positive Steady State 3.4 Discussion . . . .

4 Bistable Traveling Waves in an Epidemic Model 4.1 Introduction .

4.2 Existence of Traveling Waves 4.3 Attractivity and Uniqueness 4.4 Global Exponential Stability 4.5 Numerical Simulations

59 62 66 72

84

85 88 99 110 118

5 Spreading Speed and Traveling Waves for a Nonlocal Epidemic Model

5.1 Introduction

5.2 The Asymptotic Speed of Spread 5.3 Traveling Wave Solutions. . . . .

Bibliography

vii

121 122 124 132

141

List of Figures

3.1 The solution of Example 2 in the case of the Neumann boundary condition. The parameters of the system are as follows:

n =

(0,7r),

p

=

^q

=

^J-lj

=

^{1, d}m

=

^{0.5, (3}

=

^{0.2, d}j

=

^{0.25, T}

=

^{1.7, ¢>(t, x)}

=

1 - cos(2x). . . . . 78 3.2 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for ^T

=

1, ¢>(t, x)

=

1 - cos(2x). . . . . 78 3.3 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for ^T

=

0.3, ¢>(t, x)

=

1 - cos(2x). . . . . 79 3.4 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for ^T

=

0.3, ¢>(t, x)

=

3 - 3 cos(2x). . . . . 79 3.5 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for ^{T =} 0.3, ¢>(t, x) = 1 - cos(4x). . . . . 80 3.6 The solution of Example 2 with the same condition and parameters

as in Figure 3.1, except for ^T = 0.3, ¢>(t, x) = 5 - 5 cos( 4x). . . . 80

Vill

tion. The parameters of the system are as follows:

n =

(0, 7r),p

=

5, q

= fJ =

1, ^J1j

=

1.2, dj

=

0.25, dm

=

0.5, ^T

=

1, ¢(t, x)

=

sin x. 81 3.8 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for ^T

=

0.65, ¢(t, x)

=

sinx. . . .. 81 3.9 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for T

=

0.3, ¢(t, x)

=

sin x. . . . 82 3.10 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for ^{T =} 0.3, ¢(t, x) = 3sinx. . . . . 82 3.11 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for ^T

=

0.3, ¢(t, x)

=

1 - cos 4x. . . . 83 3.12 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for ^T

=

0.3, ¢(t, x)

=

5 - 5cos4x. . . . . 83 4.1 Illustration of the three equilibria E- ,Eo and E+.

4.2 Phase portrait of (4.2.9) . . . . 4.3 The initial function for UI component.

4.4 The initial function for ^U2 component.

4.5 The evolutionary graph of ^UI(X, t).

4.6 The evolutionary graph of ^U2(X, t).

5.1 The evolutionary graph of UI(t, x) for the solution (UI(t, x), U2(t, x)) 90 95

· 119

· 119

· 120

· 120

with initial function (5.3.19) . . . 138 5.2 The evolutionary graph of U2(t, x) for the solution (UI(t, x), U2(t, x))

with initial function (5.3.19) . . . 138

IX

3.7 The solution of Example 2 in the case of the Dirichlet boundary condi- tion. The parameters of the system are as follows:

n =

(0, w),p

=

5, q

=

j3

=

1, ^J-lj

=

1.2, dj

=

0.25, dm

=

0.5, T

=

1, ¢(t, x)

=

sin x.

3.8 The solution of Example 2 with the same condition and parameters 81

as in Figure 3.7, except for ^T

=

0.65, ¢(t,x)

=

sinx . . . 81 3.9 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for ^T

=

0.3, ¢(t, x)

=

sin x. . . . 82 3.10 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for ^T = 0.3, ¢(t, x) = 3 sin x. 82 3.11 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for ^T = 0.3, ¢(t, x) = 1 - cos 4x. 83 3.12 The solution of Example 2 with the same condition and parameters

as in Figure 3.7, except for ^T

=

0.3, ¢( t, x)

=

5 - 5 cos 4x. . . . 83 4.1 Illustration of the three equilibria E-, EO and E+. 90 4.2 Phase portrait of (4.2.9). . . 95

4.3 The initial function for ^UI component. 119

4.4 The initial function for U2 component. 119

4.5 The evolutionary graph of UI (x, t). 120

4.6 The evolutionary graph of U2(X, t). 120

5.1 The evolutionary graph of UI(t, x) for the solution (UI(t, x), U2(t, x))

with initial function (5.3.19) . . . 138 5.2 The evolutionary graph of U2(t, x) for the solution (UI(t, x), U2(t, x))

with initial function (5.3.19) . . . 138

IX

5.4 The first component of the solution (Ul(t, x), U2(t, x)) with initial function (5.3.20). . . . . 139 5.5 The second component of the solution (Ul(t, x), U2(t, x)) with initial

function (5.3.20). . . 140 5.6 The first component of the solution at some specific times. . 140

x

Chapter 1

Preliminaries

In this chapter, we present some basic theorems which will be used in this thesis.

They involve persistence theory, monotone dynamical systems, spectrum analysis, and newly developed theory for asymptotic speeds of spread and traveling waves.

1.1 Uniform Persistence

In population dynamics, uniform persistence is one of important concepts which characterize the long-term existence of species in an ecosystem. Let X be a metric space with metric d, and

f :

X --t X a continuous map. Suppose Xo is an open subset of X. Define axo := X \ X o, and Ma ^:= {x E axo: fn(x) E axo, 'in

>

O}.

Definition 1.1.1 A subset A

c

^X is said to be an attractor for f if A is nonempty, compact and invariant (f(A)

=

A), and A attracts some open neighborhood of itself.

A global attractor for f : X --t X is an attractor that attracts every point in X.

Definition 1.1.2 f is said to be uniformly persistent with respect to (Xo, aXo) if there exists TJ

>

0 such that liminfd(fn(x),aXo)

>

TJ for all x E Xo.

n-+oo

5.4 The first component of the solution (Ul(t,X),U2(t,X)) with initial

function (5.3.20). . . . . 139 5.5 The second component of the solution (Ul(t, x), U2(t, x)) with initial

function (5.3.20). . . 140 5.6 The first component of the solution at some specific times. 140

x

Chapter ¹

Preliminaries

In this chapter, we present some basic theorems which will be used in this thesis.

They involve persistence theory, monotone dynamical systems, spectrum analysis, and newly developed theory for asymptotic speeds of spread and traveling waves.

1.1 Uniform Persistence

In population dynamics, uniform persistence is one of important concepts which characterize the long-term existence of species in an ecosystem. Let X be a metric space with metric d, and

f :

X ^--7 X a continuous map. Suppose Xo is an open subset of X. Define axo := X \ X o, ^and M{) := {x E axo: fn(x) E axo, ^\In

>

O}.

Definition 1.1.1 A subset A c X is said to be an attractor for f if A is nonempty, compact and invariant (f(A)

=

A), and A attracts some open neighborhood of itself.

A global attractor for f : X --7 X is an attractor that attracts every point in X.

Definition 1.1.2 f is said to be uniformly persistent with respect to (Xo, aXo) if there exists ^'T] > 0 such that lim inf d(fn(x), aXo) > ^'T] for all x E Xo.

n-too

Theorem 1.1.1 ([94, Theorem 2.2j) Let f X --+ X be a continuous map with f(Xo) C Xo. Assume that

(1) f : X --+ X has a global attractor A;

(2) Let Aa

=

An Ma be the maximal compact invariant set of in axo. .fia UXEAa w(x) has an isolated and acyclic covering

U:=I

Mi in axo, that is, Aa ^C

U:=I

M i, where MI , M2 , . . . ,Mk are pairwise disjoint, compact and isolated invariant sets of f in axo such that each Mi is also an isolated invariant set in X, and no subset of the Mi'S forms a cycle for fa = flAa in Aa.

(3) WS(Mi) nXo

= (/)

for each 1

<

ⁱ

<

^k, where WS(Mi)

=

{x: x E X,w(x)

=I

(/) and w(x) C M i } is the stable set of Mi.

Then f is uniformly persistent with respect to (Xo, aXo).

Theorem 1.1.2 ([94, Theorem 2.3j and [63, Theorem 4.5j) Let f : X --+ X be a continuous map with f(Xo) C X o, where ^X is a closed subset of a Banach space, and Xo is a convex and relatively open subset in X. Assume that

(1) f : X --+ X is point dissipative and uniformly persistent with respect to (Xo, aXo);

(2) f is a-condensing, and fn^o is compact for some integer no

>

1.

Then f : Xo --+ Xo admits a global attractor A o, and f has a fixed point Xo ^E Ao.

For an autonomous semiflow T(t) : X --+ X, t

>

0, we can define uniform persistence by replacing fn with T(t) (see [81]). Furthermore, the continuous-time

2

Theorem 1.1.1 ([94, Theorem 2.2j) Let f X --+ X be a continuous map with f(Xo) C Xo. Assume that

(1) f : X --+ X has a global attractor A;

(2) Let Aa

=

An Ma be the maximal compact invariant set of in axo. Aa

UXEAa

w(x) has an isolated and acyclic covering

U7=1

Mi in axo, that is, Aa ^C

U7=1

M i, where M^{I ,} M2 , . . . ,Mk are pairwise disjoint, compact and isolated invariant sets of f in axo such that each Mi is also an isolated invariant set in X, and no subset of the Mi'S forms a cycle for fa = flAa in Aa.

(3) WS(Mi)

n

^Xo

⁼

^{0 for each 1}

^<

ⁱ

^<

k, where WS(Mi)

=

{x: x E X, w(x)

=I

o

^{and w(x) C} M i } is the stable set of Mi.

Then f is uniformly persistent with respect to (Xo, aXo).

Theorem 1.1.2 ([94, Theorem 2.3j and [63, Theorem 4.5j) Let f : X --+ X be a continuous map with f(Xo) C X o, where X is a closed subset of a Banach space, and Xo is a convex and relatively open subset in X. Assume that

(1) f : X --+ X is point dissipative and uniformly persistent with respect to (Xo, aXo);

(2) f is a-condensing, and fn^o is compact for some integer no

>

1.

Then f : Xo --+ Xo admits a global attractor A_o, and f has a fixed point Xo E Ao.

For an autonomous semiflow T(t) : X --+ X, t

>

0, we can define uniform persistence by replacing fn with T(t) (see [81]). Furthermore, the continuous-time

version of Theorem 1.1.1 and 1.1.2 still hold (see [81, Theorem 4.6], [94, Theorem 2.4] or [95, Theorem 1.3.7]' and [63, Theorem 4.7]).

Theorem 1.1.3 ([76, Theorem A.2j and [95, Theorem 1.3.9}) Let (Z, Z+) be an ordered Banach space with int(Z+) =1=

0

and T(t) : X -t X, t

>

0, be an autonomous semifiow with T(t)Xo C X o, t

> o.

Assume that

(1) T(t) : X -t X is point dissipative, compact for t

>

tl

>

0, and is uniformly persistent with respect to (Xo, axo);

(2) there exists t2

>

0 such that T(t₂)X_O C int(Z+) and T(t_{2 ) :} Xo -t int(Z+) 'lS

continuous.

Then, for any given e E int(Z+), there exists (3

>

⁰ such that for any compact subset B of X o, there exists to ⁼ to(B)

>

t2 such that T(t)B

>

(3e, Vt

>

to, in Z.

1.2 Monotone Dynamical Systems

Many types of equations can generate discrete- or continuous-time monotone dy- namical systems, i.e., ordered initial values imply ordered subsequences or solutions.

These types include difference, ordinary, functional and partial differential equations.

Let E be an ordered Banach space with cone P such that int(P) =1=

0.

For x, y E E, we write x

>

y if x - yEP, x

>

y if x - yEP \ {O}, and x

»

^y if x - y E int(P).

By an order interval [a, b], we mean that [a, b] = {x E E : ,a

<

x

<

b}.

Definition 1.2.1 Let U be a subset of E, and

f :

U -t U a continuous map.

The map

f

is said to be monotone if x

>

y implies that

f

(x)

> f

(y); strictly

3

version of Theorem 1.1.1 and 1.1.2 still hold (see [81, Theorem 4.6], [94, Theorem 2.4] or [95, Theorem 1.3.7], and [63, Theorem 4.7]).

Theorem 1.1.3 ([76, Theorem A.2) and [95, Theorem 1.3.9]) Let (Z, Z+) be an ordered Banach space with int(Z+)

i=

0 and T(t) : X ~ X, t

>

0, be an autonomous semiflow with T(t)Xo C X o, t

>

0. Assume that

(1) T(t) : X ~ X is point dissipative, compact for t > tl > 0, and is uniformly persistent with respect to (Xo, axo);

(2) there exists t2 >

°

such that T(t2)X^{O C} int(Z+) and T(t 2) : ^{Xo ~} int(Z+) 'ts continuous.

Then, for any given e E int(Z+), there exists (3 >

°

such that for any compact subset B of X o, there exists to ⁼ to(B)

>

t2 such that T(t)B

>

(3e, Vt

>

to, in Z.

1.2 Monotone Dynamical Systems

Many types of equations can generate discrete- or continuous-time monotone dy- namical systems, i.e., ordered initial values imply ordered subsequences or solutions.

These types include difference, ordinary, functional and partial differential equations.

Let E be an ordered Banach space with cone P such that int(P)

i= 0.

For x, y E E, we write x > y if x - YEP, x > y if x - yEP \ {a}, and x ~ y if x - y E int(P).

By an order interval [a, b], we mean that [a, b] = {x E E : ,a

<

x

<

b}.

Definition 1.2.1 Let U be a subset of E, and f : U ~ U a continuous map.

The map f is said to be monotone if x

>

y implies that f(x)

>

f(y); strictly

monotone if x

>

y implies that f (x)

>

f (y); strongly monotone if x

>

y implies that f(x)

»

^f(y)·

Definition 1.2.2 Let U be a nonempty closed and order convex set in P. A contin- uous map f : U -+ U is said to be subhomogeneous (or sublinear) if f(ax)

>

af(x) for any x E U and a E [0,1]; strictly subhomogeneous if f(ax)

>

af(x) for any x E U with x ~ 0 and a E (0,1); strongly subhomogeneous if f(ax)

»

af(x) for

any x E U with x

»

0 and a E (0,1).

Definition 1.2.3 A linear operator L on E is said to be positive if L(P) C P;

strongly positive if L(P \ {O}) C int(P). Denote by r(L) the spectral radius of L.

Theorem 1.2.1 ([93, Theorem 2.3J or [95, Theorem 2.3.4J) Let either V = [0, b]

with b ~ 0 or V = P. Assume that (1) f : V -+ V satisfies either

(i) f is monotone and strongly sublinear; or (ii) f is strongly monotone and strictly sublinear;

(2) f : V -+ V is asymptotically smooth, and every positive orbit of f in V ^2S bounded.

(3) f(O)

=

0, and the Fn3chet derivative D f(O) of f at zero is compact and strongly positive.

Then there exist threshold dynamics:

(a) If r(D f(O))

<

1, then every positive orbit in V converges to zero.

4

monotone if x > y implies that f(x) > f(y); strongly monotone if x > y implies that f(x)

»

f(y)·

Definition 1.2.2 Let U be a nonempty closed and order convex set in P. A contin- uous map f : U --+ U is said to be subhomogeneous (or sub linear) if f(ax)

>

af(x) for any x ^E U and a ^E [0,1]; strictly subhomogeneous if f(ax) > af(x) for any x E U with x

»

0 and a E (0,1); strongly subhomogeneous if f(ax)

»

af(x) for

any x E U with x» 0 and a E (0,1).

Definition 1.2.3 A linear operator L on E is said to be positive if L(P) C P;

strongly positive if L(P \ {O}) C int(P). Denote by r(L) the spectral radius of L.

Theorelll 1.2.1 ([93, Theorem 2.3) or [95, Theorem 2.3.4}) Let either V

=

[0, b]

with b

»

⁰ or V = P. Assume that (1) f : V --+ V satisfies either

(i) f is monotone and strongly sublinear; or (ii) f is strongly monotone and strictly sublinear;

(2) f : V --+ V is asymptotically smooth, and every positive orbit of f in V ~s

bounded.

(3) f(O)

=

0, and the Frechet derivative Df(O) of f at zero is compact and strongly positive.

Then there exist threshold dynamics:

(a) Ifr(Df(O))

<

1, then every positive orbit in V converges to zero.

(b) If r(D f (0))

>

1, then there exists a unique fixed point u*

»

0 in V such that every positive orbit in V \ {O} converges to u*.

For an autonomous semiflow T(t) on E, we can define monotonicity and sub- linearity in a similar ways. Moreover, there exists a continuous-time version of Theorem 1.2.1. [96, Theorem 3.2] is the version for delay differential equations, and [96, Corollary 3.2] is the version for ordinary differential equations.

Theorem 1.2.2 ([95, Theorem 2.2.4}) Let U be a closed convex subset of an ordered Banach space X, and <t>(t) : U ~ U be a monotone semiflow. Assume that there exists a monotone homeomorphism h from

[0, 1]

onto a subset of U such that

(1) For each s E [0,1]' h(s) is a stable equilibrium for <t>(t) : U ~ U;

(2) Each orbit of <t>(t) in [h(O), h(l)]x is precompact;

(3) One of the following two properties holds:

(3a) If h(so) <x w(¢) for some So E [0,1) and ¢ E [h(O), h(l)]x, then there exists Sl E (so, 1) such that h(Sl) <x w(¢);

(3b) If w(¢) <x h(rd for some r1 E (0,1] and ¢ E [h(O), h(l)]x, then there exists ro E (0, r1) such that w(¢) <x h(ro).

Then for any precompact orbit ry+(¢o) of <t>(t) in U with w(¢o)

n

[h(O), h(l)]x

=I-

^(/J,

there exists s* E [0,1] such thatw(¢o)

=

h(s*).

The following attractivity theorem is due to M. W. Hirsch ([48]), and is a powerful tool to prove the global attractivity of a unique equilibrium.

5

(b) If r(D f (0)) > 1, then there exists a unique fixed point u* ~ 0 in V such that every positive orbit in V \ {O} converges to u*.

For an autonomous semiflow T(t) on E, we can define monotonicity and sub- linearity in a similar ways. Moreover, there exists a continuous-time version of Theorem 1.2.1. [96, Theorem 3.2] is the version for delay differential equations, and [96, Corollary 3.2] is the version for ordinary differential equations.

Theorem 1.2.2 ([95, Theorem 2.2.4}) Let U be a closed convex subset of an ordered Banach space X, and <I>(t) : U ---+ U be a monotone semifiow. Assume that there exists a monotone homeomorphism h from [0,1] onto a subset of U such that

(1) For each s E [0,1], h(s) is a stable equilibrium for <I>(t) : U ---+ U;

(2) Each orbit of <I>(t) in [h(O), h(I)]x is precompact;

(3) One of the following two properties holds:

(3a) If h(so) <x w(¢) for some So E [0,1) and ¢ E [h(O), h(I)]x, then there exists Sl E (so, 1) such that h(Sl) <x w(¢);

(3b) If w(¢) <x h(rd for some r1 E (0,1] and ¢ ^E [h(O), h(I)]x, then there exists ro E (0, r1) such that w( ¢) <x h(ro).

Then for any precompact orbit 7+(¢0) of <I>(t) in U with w(¢o)

n

[h(O), h(I)]x -=I-

0,

there exists s* E [0,1] such that w(¢o)

=

h(s*).

The following attractivity theorem is due to M. W. Hirsch ([48]), and is a powerful tool to prove the global attractivity of a unique equilibrium.

Theorem 1.2.3 ([48, Theorem 3.3j) Let T(t) : E ^-7 E be a monotone semifiow.

Assume that T(t) admits an attractor K such that K contains only one equilibrium

X*. Then every trajectory attracted to K converges to ^X*.

In the following, we introduce two theorems about competitive systems on or- dered Banach spaces, which are one of the main tools in Chapter 2. For i = 1,2, let Xi be ordered Banach spaces with positive cones xt, where int(Xt) =1=

0.

^Let X = Xl ^X X 2, X+ ⁼ x i x xi, and K = x i x (-Xi). Denote by <K the order on X defined by K. The following hypotheses ([50]) are meant to capture the essence of competition between two adequate competitors:

(AI) f : X+ -7 X+ is strictly monotone with respect to <K, and is order compact in the sense that f([O, Xl] x [0, X2]) is precompact in X for every (Xl, X2) E X+.

(A2) 0 is a repelling fixed point of

f

in the sense that there exists a neighborhood Ua of 0 in X+ such that for each X E Ua with ^{X =1=} 0, there is an integer n such that fn(x)

tf-

Ua.

(A3) f(Xi x {O})

c

x i x {O}, and there exists Xl E int(Xi) such that f((Xl' 0))

=

(Xl,O) and the omega limit set W((Xl'O)) of the orbit fn(Xl, 0) is (Xl,O) for every Xl E Xi \ {O}. The symmetric conditions hold for f on {O} x X 2, ^and the fixed point is denoted by (0, X2).

(A4) If x, y E X+ satisfy X <K Y and either X or y belongs to int(X+), then f(x) ~K f(y)· If X

=

^{(Xl, X2) E X+ with Xi =1= 0, i}

=

^{1,2, then f(x)}

» o.

Denote the "boundary" fixed points of f by Ea = (0,0), El = (Xl, 0), E2 = (0, X2).

Let I = [E2' El]K := {x EX: E2 <K X <K E l }. Obviously, I = [0, Xl] x [0, X2].

6

Theorem 1.2.3 ([48, Theorem 3.3j) Let T(t) : E --+ E be a monotone semifiow.

Assume that T(t) admits an attractor K such that K contains only one equilibrium

x*. Then every trajectory attracted to K converges to X*.

In the following, we introduce two theorems about competitive systems on or- dered Banach spaces, which are one of the main tools in Chapter 2. For i = 1,2, let Xi be ordered Banach spaces with positive cones xt, where int(Xt)

#-

0. Let X ⁼ Xl ^X X2 , X+ ⁼ x t x Xi, and K = x t x (-Xi). Denote by <K the order on X defined by K. The following hypotheses ([50]) are meant to capture the essence of competition between two adequate competitors:

(AI) f : X+ --+ X+ is strictly monotone with respect to <K, and is order compact in the sense that f([O, Xl] x [0, X2]) is precompact in X for every (Xl, X2) ^E X+.

(A2)

°

is a repelling fixed point of

f

in the sense that there exists a neighborhood U_{o of}

°

^{in X+} such that for each X E U_{o with} X

#-

0, there is an integer n such that fn(x) ¢ Uo.

(A3) f(xt x {O}) c x t x {O}, and there exists Xl E int(Xt) such that f((Xl' 0))

=

(Xl,O) and the omega limit set W((Xl' 0)) of the orbit fn(Xl,O) is (Xl,O) for every ^Xl E x t \ {O}. The symmetric conditions hold for

f

on {O} x X2 , and the fixed point is denoted by (0, X2).

(A4) If x, y E X+ satisfy X <K Y and either X or y belongs to int(X+), then f(x) 4:;.K fey). If X = (Xl, X2) E X+ with Xi

#-

0, i = 1,2, then f(x)

»

0.

Denote the "boundary" fixed points of f by Eo ⁼ (0,0), El = (Xl, 0), E2 ⁼ (0, X2).

Let I = [E2' El]K ^:= {x EX: E2 <K X <K E l }. Obviously, I = [0, Xl] x [0, X2].

The following result ([50]) says that for a competitive system, either there is a positive fixed point of

f,

representing coexistence of the two populations, or one population drives the other to extinction.

Theorem 1.2.4 ([50, Theorem A.l}) Let (Al)-(A4) hold. Then the omega limit set of every orbit in X+ is contained in I, and exactly one of the following holds:

(a) There exists a positive fixed point E* of f in I;

(b) w(x)

=

EI for every x

=

(Xl, X2) E I with Xi

i-

0, i

=

1,2;

(c) w(x)

=

E2 for every X

=

(Xl, X2) E I with Xi

i-

0, i

=

1,2.

Finally, if (b) or (c) holds and X = (Xl, X2) E X+ \ I with Xi

i-

0, i = 1,2, then either w(x)

=

El or w(x)

=

E_{2 .}

Theorem 1.2.5 ([95, Theorem 2.4.2}) Let (Al)-(A4) hold and assume that El and E2 are isolated fixed points of

f.

Let WS (Ei) be the stable set of Ei for f.

If WS(Ei)

n

^int(X+)

^{= (/),}

ⁱ

⁼

^1,2, then there exist positive fixed points E** <K E*

of f such that w(x)

=

^E* for every X

=

^{(Xl, X2)} satisfying E* <K X <K EI and ' X2

i-

^{0, w(x)}

=

^E** for every X

=

^{(Xl, X2)} satisfying E2 <K X <K E** and Xl

i-

^0,

and the order interval [E**, E*] K attracts any point in (Xi \ {o}) x (Xi \ {o} ).

1.3 Essential Spectrum

This section presents some results about essential spectrum of certain ordinary dif- ferential operators obtained in [46, Page 136-142].

7

The following result ([50]) says that for a competitive system, either there is a positive fixed point of

f,

representing coexistence of the two populations, or one population drives the other to extinction.

Theorem 1.2.4 ([50, Theorem A.lj) Let (Al)-(A4) hold. Then the omega limit set of every orbit in X+ is contained in I, and exactly one of the following holds:

(a) There exists a positive fixed point E* of f in I;

(b) w(x)

=

EI for every x

=

(Xl, X2) E I with Xi

#-

0, i

=

1,2;

(c) w(x)

=

E2 for every X

=

(Xl, X2) E I with Xi

#-

0, i

=

1,2.

Finally, if (b) or (c) holds and X

=

(Xl, X2) E X+ \ I with Xi

#-

^{0, i}

=

1,2, then either w(x)

=

EI or w(x)

=

_{E 2.}

Theorem 1.2.5 ([95, Theorem 2.4.2j) Let (Al)-(A4) hold and assume that EI and E2 are isolated fixed points of

f.

Let WS (Ei) be the stable set of Ei for f.

If WS(Ei)

n

^int(X+)

= 0,

i

=

^1,2, then there exist positive fixed points E** <K E*

of f such that w(x) = E* for every X = (Xl, X2) satisfying E* <K X <K EI and ' X2

#-

0, w(x)

=

^E** for every X

=

^{(Xl, X2)} satisfying E2 <K X <K E** and Xl

#-

0, and the order interval [E**, E*]K attracts any point in

(xt \

{o}) x (Xi \ {o}).

1.3 Essential SpectruIIl

This section presents some results about essential spectrum of certain ordinary dif- ferential operators obtained in [46, Page 136-142].

Definition 1.3.1 If L is a linear operator in a Banach space, a normal point for L is any complex number which is in the resolvent set, or is an isolated eigenvalue of L with finite multiplicity. Any other complex number is in the essential spectrum.

Denote the resolvent set and spectrum of L by p(L) and a(L), respectively.

Theorem 1.3.1 ([46, Lemma 2j) Suppose the matrices A+(A), A_(A) are analytic functions of A E C, the complex number set. Let

S±

=

{A: A±(A) has an imaginary eigenvalue }.

Let A(x, A)

=

A+(A) for x > 0, A_(A) for x < 0, and define the differential operator L(A)U

=

d~u

+

A(·, A)U in Co(lR), the continuous function set, or Cuni/(JR), the uniformly continuous function set; we may consider L(A) as closed and densely defined. Then if G is any open connected set in C \ (S+

U

S_), either

(i) 0 E a(L(A)) for all A in G, or

(ii) 0 E p( L( A)) for all A in G except at isolated points, the exceptional points are

poles of L(A)-l of finite order .

. Also, 0 E a(L(A)) whenever A E S+

U

^S_.

Theorem 1.3.2 ([46, Theorem A.lj) Suppose X is a Banach space, T : D(T)

c

X -t X is a closed linear operator, S : D(S)

c

X -t X is linear with D(T)

c

D(S) and S(AoI - T)-l is compact for some AO' Let U be an open connected set in C consisting entirely of normal points of T; then either U consists entirely of normal points of T

+

S, or entirely of eigenvalues of T

+

S.

8

Definition 1.3.1 If L is a linear operator in a Banach space, a normal point for L is any complex number which is in the resolvent set, or is an isolated eigenvalue of L with finite multiplicity. Any other complex number is in the essential spectrum.

Denote the resolvent set and spectrum of L by p(L) and CJ(L), respectively.

Theorem 1.3.1 ([46, Lemma 2j) Suppose the matrices A+(,\), A_('\) are analytic functions of ,\ ^E C, the complex number set. Let

S±

= {'\:

A±('\) has an imaginary eigenvalue }.

Let A(x,'\) = A+('\) for x > 0, A_('\) for x < 0, and define the differential operator L('\)u = d~ u

+

A(·, '\)u in Co (JR), the continuous function set, or Cunij(JR), the uniformly continuous function set; we may consider L('\) as closed and densely defined. Then if G is any open connected set in C \ (S+

U

S_), either

(i) 0 E CJ(L('\)) for all ,\ in G, or

(ii) 0 E p(L('\)) for all ,\ in G except at isolated points, the exceptional points are poles of L(,\)-l of finite order.

Also, 0 E CJ(L('\)) whenever'\ E S+

U

S_.

Theorem 1.3.2 ([46, Theorem A.i]) Suppose X is a Banach space, T : D(T) c X ~ X is a closed linear operator, S : D(S) c X ~ X is linear with D(T) c D(S) and S('\oI - T)-l is compact for some '\0. Let U be an open connected set in C consisting entirely of normal points of T; then either U consists entirely of normal points of T

+

S, or entirely of eigenvalues of T

+

S.

1.4 Spreading Speeds and Traveling Waves

The theory of asymptotic speeds of spread and traveling waves, developed in [8, 7, 9, 26, 28, 27, 79, 80, 90], has been recently generalized to a class of scalar nonlinear integral equations in [83]. In this section, we present some results obtained in [83].

Definition 1.4.1 A number ^c* >

°

is called the asymptotic speed of spread for a function u :

Il4

x IRn ---+

Il4

if lim u( t, x)

= °

for each c > c*, and if there

t--+oo,lxl~ct

exists some

u

>

°

such that lim u(t, x)

= u

for each c E (0, c*).

t--+oo,lxl~ct

Consider an integral equation

u(t, x)

=

uo(t, x)

+ t J.

F(u(t - s, x - y), s, y)dyds,

Jo ^]Rn (1.4.1)

where F : IR~ x IRn ---+ IR is continuous in u and Borel measurable in (s, y), and

Uo :

Il4

x IRn ---+

Il4

is Borel measurable and bounded. Assume that (B) There exists a function k :

Il4

x IRn ---+

Il4

such that

(B1) k* := Jo^oo J]Rn k(s, x)dxds

<

00;

(B2)

° ^<

F(u, s, x)

<

uk(s, x), \/u, s > 0, x E IRn;

(B3) For every compact interval I in (0,00), there exists some c >

°

such that F(u, s, x)

>

ck(s, x), \/u E I, s

>

0, x E IRn;

(B4) For every c

>

0, there exists some c5 >

°

such that F(u, s, x)

>

(1 - c)uk(s, x), \/u E [0, c5], s

>

0, x E IRn;

(B5) For every w > 0, there exists some A >

°

such that

IF(u, s, x) - F(v, s, x)1 < Alu - vlk(s, x), \/u, v E [0, w], s

>

0, x E IRn.

10

The following proposition shows that the existence, uniqueness and some properties of solutions to equation (1.4.1).

Proposition 1.4.1 ([83, Proposition 2.1}) If assumptions (B) hold, then for every

Borel measurable, nonnegative and bounded function uo(t, x), there exists a unique Borel measurable solution u :

114

^x IRn -+

114

^of (1.4.1), and u is bounded on

[0, r] x IRn for every r > O. Furthermore, the following statements hold.

(a) The solution u is bounded if there exist C1, C2 > 0 such that C1 k* < 1 and F(u, s, x)

<

^(C2

+

^c1u)k(s, x), Vu, s

>

0, x E IRn.

(b) If r > 0 and lim Uo (t, x)

=

0 uniformly for t E [0, r], then the solution u has Ixl--+oo

the same property.

To obtain some more properties for equation (1.4.1), we have to make some assumptions on k.

( C) k:

114

^x IRn -+

114

is a Borel measurable function such that (C1) k* := fo^oo flRn k(s, y)dyds E (1,00);

(C2) There exists some ,\0 > 0 such that fo^oo flRn e^AoY1 k(s, y)dyds < 00;

(C3) There exist numbers 0"2

>

^0"1

>

0, p

>

0 such that k(s, x)

>

0, Vs E

(C4) k is isotropic.

Here, a function

f :

[0,00) x IRn -+ IR is said to be isotropic if for almost all s > 0,

f(s, x)

=

f(s, y) whenever

Ixl = Iyl.

For a fixed Z E IRn with

IZI =

1, define the

1.4 Spreading Speeds and Traveling "Waves

The theory of asymptotic speeds of spread and traveling waves, developed in [8, 7, 9, 26, 28, 27, 79, 80, 90], has been recently generalized to a class of scalar nonlinear integral equations in [83]. In this section, we present some results obtained in [83].

Definition 1.4.1 A number c* >

°

is called the asymptotic speed of spread for a function u :

ll4

x JRn --+

ll4

if lim u( t, x) =

°

for each c > c*, and if there

t-+oo,lxl2':ct

exists some

u

>

°

such that lim u(t, x)

= u

for each c E (0, c*).

t-+oo,lxl:::;ct

Consider an integral equation

u(t, x) = uo(t, x)

+ it ^r

F(u(t - s, x - y), s, y)dyds,

o J'Rⁿ (1.4.1)

where F : JR~ x JRn --+ JR is continuous in u and Borel measurable in (s, y), and Uo :

ll4

x JRn --+

ll4

is Borel measurable and bounded. Assume that

(B) There exists a function k :

ll4

x JRn --+

ll4

such that (B1) k* := Jo^oo J'Rn k(s, x)dxds

<

00;

(B2)

° ^<

F(u, s, x)

<

uk(s, x), Vu, s

>

0, x ^E JRn;

(B3) For every compact interval I in (0,00), there exists some c >

°

such that F(u, s, x)

>

ck(s, x), Vu E I, s

>

0, x E JRn;

(B4) For every c

>

0, there exists some 5

> °

such that F(u, s, x)

>

(1 - c)uk(s, x), Vu E [0,5], s

>

0, x E JRn;

(B5) For every w > 0, there exists some A >

°

such that

IF(u, s, x) - F(v, s, x)1

<

Alu - vlk(s, x), Vu, v E [0, w], s

>

0, x E JRn .

10

The following proposition shows that the existence, uniqueness and some properties of solutions to equation (1.4.1).

Proposition 1.4.1 ([83, Proposition 2.1j) If assumptions (B) hold, then for every Borel measurable, nonnegative and bounded function Uo (t, x), there exists a unique Borel measurable solution u :

114

^{x ]Rn -+}

114

of (1.4.1), and u is bounded on [0, r] x ]Rn for every r > ^0. Furthermore, the following statements hold.

(a) The solution u is bounded if there exist Cl, C2 >

°

such that clk* < 1 and F(u, s, x)

<

(C2

+

clu)k(s, x), \lu, s

>

0, x E ]Rn.

(b) If r >

°

and lim uo(t, x) =

°

uniformly for t ^E [0, r], then the solution u has Ixl-+oo

the same property.

To obtain some more properties for equation (1.4.1), we have to make some assumptions on k.

(C) k:

114

x ]Rn -+

114

is a Borel measurable function such that (C1) k* :=

Iooo IlRn

k(s, y)dyds E (1,00);

(C2) There exists some AO >

°

such that

Iooo IlRn

e)..<>Yl k(s, y)dyds < 00;

(C3) There exist numbers ^0"2 > ^0"1 > ^0, p >

°

such that k(s, x) > ^0, \Is E

(C4) k is isotropic.

Here, a function

f :

[0,00) x ]Rn -+ ]R is said to be isotropic if for almost all s > 0, /(3, x) = /(3, y) whenever

Ixl

=

Iyl.

For a fixed Z E ]Rn with

IZI

= 1, define the

transform

K(e,'x):=

t)Q!.

e-)..(cs-z·Y)k(s, y)dyds, \:Ie

>

0, ,x

>

0,

io

^]Rn

where . means the usual inner product on ]Rn. Suppose that k is isotropic. Since for any Z E ]Rn with

IZI =

1, there exists an orthogonal matrix A with AZ

=

-el, where el is the first canonical basis vector of ^]Rn, there holds

K(e,'x)

= 1°O!.

e-)..(cs+yd k(s, y)dyds, o ^]Rn

where Yl is the first coordinate of y. If (C) holds, then, for every e > 0, there exists some 'x~(e) E (0,00] such that K(e,'x) < 00 for ,x E [0, 'x~(e)) and K(e,'x)

=

00 for

,x

>

'x~(e) ([79, Lemma 3.7]). Let e* := inf{e

>

0 : K(e,'x) < 1 for some ,x > O}.

The following lemma shows the existence of e*.

Lemma 1.4.1 ([83, Proposition 2.3j) Let (C) hold and assume that lim inf K(e,'x)

2

)..,l').." (c)

k* for every e >

o.

Then there exists a unique ,x * E (O,,x~ (e*)) such that K( c*, ,x *)

=

1 and K(e*,'x) > 1 for ,x

i=

^,X*. Moreover, e* and ,X* are uniquely determined as the solutions of the system K(e,'x)

=

1, d~K(e,'x)

= o.

The function Uo is said to be admissible if for every e,'x > 0 with K(e,'x) < 1, there exits some "'( > 0 such that uo(t, x) < "'(e)..(ct-1xl), \:It

2

0, x E ^]Rn. The following theorems show that e* is the asymptotic speed of spread for solutions of (1.4.1).

Theorem 1.4.1 ([83, Theorem 2.1j) Let (B) and (C) hold. then for every admis- sible Uo, the unique solution u(t, x) of {1.4.1} satisfies lim u(t, x)

=

0 for each

t--too,lxl~ct

e> e*.